The document discusses Heaviside's unit step function and its applications. The unit step function u(t) is defined as having a value of 0 for negative time and 1 for positive time. A shifted unit step function u(t-a) has a value of 0 up until time t=a and then 1. Heaviside's functions can be used to represent signals and waveforms that switch on or off at certain times. They are commonly used in control theory, signal processing, and calculating currents in electric circuits when switches are turned on or off.

1. Laplace Transform Of
Heaviside’s Unit Step Function

2. Laplace Transform Of Heaviside’s
Unit Step Function
 Definition: The unit step function is denoted as u(t) or H(t) and is defined as:
That is, u is a function of time t, and u has value zero when time is negative and
value one when time is positive. Graphically it can be represented as :-

4. Shifted Unit Step Function:
 In many circuits, waveforms are applied at specified intervals other than t = 0.
 Such a function may be described using the shifted /delayed unit step function.
 A function which has value 0 up to the time t = a and thereafter has value 1 is known as
shifted unit step function and is written as:
 Graphically it can be represented as :-

6. Representation Of A Function
Using Heaviside’s Functions:
 It is more convenient to represent a function with the help of unit step
function
 A function f(t) can be represented in different ways using Heaviside’s
function.
 i. F(t).H(t)
 ii. F(t).H(t – a)
 iii. F(t – a).H(t)
 iv. F(t – a).H(t – b)
 v. F(t) from t = a to t = b

7. Applications of Heaviside’s Unit
Step Function:
Where do we use it?
 The function is commonly used in the mathematics of control theory and signal
processing.
 Heaviside’s unit step function represents unit output of a system with possible time
lead or lag.
 It is used to calculate currents when electric circuit is switches on.
 It represents a signal that switches on at a specified time stays switched on
indefinitely.

8. How do we use it?
 Heaviside functions can only take values 0 or 1, but we can also use them to get
other kinds of switches.
 Example: 4uc(t) is a switch that is off until t = c and then turns on and takes a value
4.
 Now, suppose we want a switch that is on (with a value 1) and then turns off at t = c.
 We can represent this by 1 – uc (t) = {1 – 0 = 1} ; if 𝑡 < 𝑐
= {1 – 1 = 0} ; if 𝑡 ≥ 𝑐
Heaviside’s Unit Step Function:

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